6.2 Properties of Power Series

Power series let you treat functions as infinite sums and then manipulate them with familiar algebraic and calculus operations. Understanding how these operations work, and what happens to the interval of convergence when you apply them, is central to working with power series throughout Calculus II.

Properties of Power Series

Operations on power series

You can add, subtract, and scalar-multiply power series much like polynomials, working term by term with the coefficients.

Addition and subtraction: Add or subtract the corresponding coefficients of like powers of $x$:

$\sum_{n=0}^{\infty} a_nx^n \pm \sum_{n=0}^{\infty} b_nx^n = \sum_{n=0}^{\infty} (a_n \pm b_n)x^n$

The resulting series converges on the intersection of the two original intervals of convergence. If both series share the same interval, the result does too.

Scalar multiplication: Multiply every coefficient by the constant $c$:

$c\sum_{n=0}^{\infty} a_nx^n = \sum_{n=0}^{\infty} (ca_n)x^n$

The interval of convergence stays the same.

These term-by-term operations are valid because power series converge uniformly on any closed subinterval strictly inside their interval of convergence.

Variable substitution in power series

Replacing the variable with a new expression generates a new power series. For example, substituting $x^2$ for $x$ in the geometric series gives:

$\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \quad \longrightarrow \quad \frac{1}{1-x^2} = \sum_{n=0}^{\infty} x^{2n}$

To find the new interval of convergence, apply the original convergence condition to the substituted expression. The geometric series converges for $|x| < 1$, so the substituted series converges when $|x^2| < 1$, which is still $|x| < 1$.

Multiplying by a power of $x$ shifts the exponent on every term:

$x^2 \cdot \sum_{n=0}^{\infty} a_nx^n = \sum_{n=0}^{\infty} a_nx^{n+2}$

This doesn't change the radius of convergence (multiplying by $x^2$ doesn't affect the ratio or root test), though you should always double-check endpoints.

Multiplication of power series

Multiplying two power series uses the Cauchy product. The idea is the same as distributing two polynomials and collecting like powers of $x$:

$\left(\sum_{n=0}^{\infty} a_nx^n\right)\left(\sum_{n=0}^{\infty} b_nx^n\right) = \sum_{n=0}^{\infty} c_n x^n$

where the coefficient of the $n$-th term is:

$c_n = \sum_{k=0}^{n} a_k \, b_{n-k}$

Each $c_n$ comes from multiplying all pairs of terms whose exponents add up to $n$, then summing those products. The resulting series converges on at least the intersection of the two original intervals of convergence.

Differentiation and integration of series

One of the most powerful properties of power series: you can differentiate and integrate them term by term, just as you would a polynomial.

Differentiation: Differentiate each term individually and adjust the starting index (the $n=0$ constant term drops out):

$\frac{d}{dx}\sum_{n=0}^{\infty} a_nx^n = \sum_{n=1}^{\infty} n\,a_n\,x^{n-1}$

The radius of convergence stays the same, though convergence at the endpoints may change. For instance, a series that converges at an endpoint might diverge there after differentiation.

Integration: Integrate each term and add a constant of integration:

$\int \sum_{n=0}^{\infty} a_nx^n\, dx = \sum_{n=0}^{\infty} \frac{a_n}{n+1}\,x^{n+1} + C$

Again, the radius of convergence is preserved. Integration can actually improve endpoint behavior: a series that diverges at an endpoint might converge there after integration. The constant $C$ is usually determined by an initial condition or set to 0 when none is given.

Advanced concepts in power series

• A function is called analytic at a point if it equals its power series representation in some neighborhood of that point. Most functions you encounter in this course ($e^x$, $\sin x$, $\ln(1+x)$, etc.) are analytic on their intervals of convergence.
• Abel's theorem addresses what happens at the boundary: if a power series converges at an endpoint of its interval, then the sum of the series at that point equals the limit of the function as you approach the endpoint from inside the interval. This is useful for evaluating series at endpoints where convergence is conditional.
• Laurent series and complex analysis extend these ideas beyond Calc II, allowing negative powers of the variable and convergence in the complex plane.